Abstract
In this paper, we devise improved approximation algorithms for the Min–Max Rural Postmen Cover Problem (RuralPostCover) and the Min–Max Chinese Postmen Cover Problem (ChinesePostCover), which are natural extensions of the classical Rural Postman Problem and the Chinese Postman Problem where multiple postmen are available. These results are based on some key observations, a new approach to derive closed walks from (open) walks and an efficient postmen allocation procedure in the literature. As an application of the algorithm for RuralPostCover, we give the first constant-factor approximation algorithms for the Min–Max Subtree Cover Problem (SubtreeCover) and its generalization, called the Min–Max Steiner Tree Cover Problem with Vertex Weights (SteinerTreeCover), using simple approximation preserving reductions. Moreover, we devise specialized algorithms for SteinerTreeCover (SubtreeCover) with better approximation ratios.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A parallel edge \(\{u,v\}\) is subdivided into two edges \(\{u,v'\}\), \(\{v',v\}\) by introducing a new vertex \(v'\). The length of \(\{u,v'\}\) is the same as the original \(\{u,v\}\) while the length of \(\{v',v\}\) is zero.
If H is a tree graph, the edge-disjoint connected subgraphs \(H_1,\ldots ,H_k\) in Lemma 11 are actually subtrees of H.
References
Applegate D, Cook W, Dash S, Rohe A (2012) Solution of a min–max vehicle routing problem. INFORMS J Comput 14:132–143
Arkin EM, Hassin R, Levin A (2006) Approximations for minimum and min–max vehicle routing problems. J Algorithms 59:1–18
Averbakh I, Berman O (1997) \((p-1)/(p+1)\)-approximate algorithms for \(p\)-traveling salesmen problems on a tree with minmax objective. Discret Appl Math 75:201–216
Byrka J, Grandoni F, Rothvob T, Sanita L (2013) Steiner tree approximation via iterative randomized rounding. J ACM 60(1):6
Corberan A, Laporte G (eds) (2015) Arc routing: problems, methods, and applications. SIAM, Philadelphia
Even G, Garg N, Koemann J, Ravi R, Sinha A (2004) Min–max tree covers of graphs. Oper Res Lett 32:309–315
Gao X, Fan J, Wu F, Chen G (2018) Approximation algorithms for sweep coverage problem with multiple mobile sensors. IEEE/ACM Trans Network 26(2):990–1003
Guan M (1962) Graphic programming using odd and even points. Chinese Math 1:273–277
Gutin G, Muciaccia G, Yeo A (2013) Parameterized complexity of \(k\)-Chinese postman problem. Theoret Comput Sci 513:124–128
Khani MR, Salavatipour MR (2014) Improved approximation algorithms for min–max tree cover and bounded tree cover problems. Algorithmica 69:443–460
Korte B, Vygen J (2018) Combinatorial optimization: theory and algorithms, 6th edn. Springer, Berlin
Lawler EL (1976) Combinatorial optimization: networks and matroids. Holt, Rinehart and Winston, New York
Mao Y, Yu W, Liu Z, Xiong J (2019) Approximation algorithms for some minimum postmen cover problems. Lect Notes Comput Sci 11949:375–386
Megiddo N (1983) Applying parallel computation algorithms in the design of serial algorithms. J ACM 30(4):852–865
Nagamochi H, Okada K (2003) Polynomial time 2-approximation algorithms for the minmax subtree cover problem. Lect Notes Comput Sci 2906:138–147
Nagamochi H, Okada K (2004) A faster 2-approximation algorithm for the minmax p-traveling salesmen problem on a tree. Discret Appl Math 140:103–114
Orloff CS (1974) A fundamental problem in vehicle routing. Networks 4:35–64
Pearn WL (1994) Solvable cases of the \(k\)-person Chinese postman problem. Oper Res Lett 16:241–244
Quirion-Blais Q, Langevin A, Lehuede F, Peton O, Trepanier M (2017) Solving the large-scale min–max \(k\)-rural postman problem for snow plowing. Networks 70:195–215
van Bevern R, Hartung S, Nichterlein A, Sorge M (2014) Constant-factor approximations for Capacitated Arc Routing without triangle inequality. Oper Res Lett 42:290–292
Willemse EJ, Joubert JW (2012) Applying min–max \(k\) postmen problems to the routing of security guards. J Oper Res Soc 63:245–260
Xu Z, Wen Q (2010) Approximation hardness of min–max tree covers. Oper Res Lett 38:169–173
Xu L, Xu Z, Xu D (2013) Exact and approximation algorithms for the min–max \(k\)-traveling salesmen problem on a tree. Eur J Oper Res 227:284–292
Yu W, Liu Z, Bao X (2021) Approximation algorithms for some min–max postmen cover problems. Ann Oper Res 300:267–287
Acknowledgements
The authors are very grateful to the anonymous referees and the editors for their insightful comments which greatly improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research is supported by the National Natural Science Foundation of China under Grant numbers 11671135, 11871213 and the Natural Science Foundation of Shanghai under Grant Number 19ZR1411800.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yu, W. Improved approximation algorithms for some min–max postmen cover problems with applications to the min–max subtree cover. Math Meth Oper Res 97, 135–157 (2023). https://doi.org/10.1007/s00186-022-00807-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-022-00807-8